Electrical power systems are continually experiencing disturbances. Of primary concern to the power industry are those disturbances classified as event disturbances, which include generator outages, short-circuits caused by lightning, sudden large load changes, or a combination of such events. It is of critical importance to be able to predict the ability of the power system to reach an acceptable steady-state operating environment after an event disturbance. This is the area of concern in system transient stability analysis.
The present invention relates to methods for analyzing power systems and, more particularly, to an on-line method for determining power system transient stability.
An event disturbance usually leads to a change in the configuration of the power system. Load disturbances are the small random fluctuations in load demands. The system usually remains unchanged after load disturbances.
In a power system the transient stability of concern is the synchronization of the phase and rotational velocity of the generators.
Power system operation is divided into three operating phases or operational configurations: pre-fault, fault-on, and post-fault. Transient stability is the study of whether the post-fault system will converge (tend) toward an acceptable steady state as time increases.
Consider the following possible scenario. An event disturbance, such as lightning, causes a short circuit in a power system. The surge in current demand caused by the short circuit causes a protective circuit breaker to trip, disconnecting the short circuit from the system. The sudden change in the load upon the generators causes at least one of them to overspeed, increasing the voltage on the system. If the post-fault condition becomes stable, the overspeeding generator will be gradually brought down to the same speed as the other generators in the system. If the post-fault conditioning will not stabilize, however, other circuit breakers could open in a chain reaction. In one worst case scenario, a significant amount of the power system generators may go off-line to protect themselves. The subsequent reduction in the service to utility customers, and the time required to reconnect the various generators and bring them back on-line, could be very costly and inconvenient. At present there is no convenient and reliable way of predicting when such an unstable condition is imminent.
Transient stability analysis is routinely performed by a utility company during system planning. The industry standard for transient stability varies from region to region, but in general requires the ability of the system to withstand severe event disturbances, such as a three-phase fault condition in close proximity to a generator with a stuck breaker.
Until recently transient stability analysis has been performed by utilities exclusively by means of the numerical integration of nonlinear differential equations describing the fault-on condition and the post-fault condition. Computer programs that can handle 2000 buses and 300 generators with detailed models are available, but may take several hours of computer time to run. Since the time interval of interest in transient stability analysis is between a few milliseconds and a few seconds, simulation for on-line security analysis which require a fast and reliable method has not been feasible, until this invention.
Mathematically the problem of transient stability in accordance with this invention can be defined by three sets of differential equations, one each for the pre-fault, fault-on, and the post-fault conditions: EQU x(t)=f.sub.1 (x(t)), -.infin.&lt;t&lt;t.sub.F EQU x(t)=f.sub.F (x(t)), t.sub.F .ltoreq.t&lt;t.sub.p EQU x(t)=f(x(t)), t.sub.p .ltoreq.t&lt;.infin. (1)
where t.sub.f is the time of fault occurrence and t.sub.p the time of fault clearance.
Since the power system varies with time a value can be calculated that is indicative of the state of a system at any given moment, which in turn can be used to determine if the system will be stable at a future point in time.
The state or condition of a system is represented by a set of numbers. These values, combined with input functions, are introduced into the equations describing the dynamics of the power system. The solution to the equations provides a prediction of the future state and output of the system. The state or condition of a system is expressed as a function of time.
All the states of a system are said to exist in state space. The path the state or condition of a system follows as time progresses is known as a "trajectory".
One of the most important determinations is whether the post-fault trajectory of the system will approach a stable equilibrium point. This will determine transient stability. Around each stable equilibrium point (SEP) is a region of stability. Any point within this stability region will approach the SEP with respect to time.
At the time a fault occurs, the system is at an SEP. While the fault is occurring, the system begins a fault trajectory out of the region of stability. If the fault is cleared before the trajectory leaves the stability region, then the post-fault condition will be stable. The time between the fault occurrence and the time when the post-fault trajectory leaves the stability region is defined by t.sub.c, the critical time.
For a power system having N generators, the classical model of the dynamics of the i.sup.th generator is governed by the equations: ##EQU1## Equations 2-a and 2-b represent rotor mechanics; .delta..sub.i, w.sub.i are the respective angle and speed deviations (from a synchronous reference) of the ith rotor; M.sub.i and D.sub.i are respective inertia and damping constants of the ith generation; P.sub.i.sup.m is the constant mechanical input power; and P.sub.i.sup.e is determined by the electrical network. The i.sup.th generator is configured as a voltage source of constant magnitude E.sub.i and phase .sigma..sub.i driving (through a transient reactance) an electric network consisting of transmission lines and loads, configured as constant impedances. Viewed as an N port driven by the voltage sources E.sub.i &lt;.delta..sub.i, this network is described by the reduced complex admittance matrix Y with coefficients Y.sup.i &lt;.phi..sub.ij (see FIG. 1). The real, positive diagonal terms G.sub.ij of Y are separated as indicated. Equation 2-c gives the real power delivered to the electric network by the ith generator. In this N generator system, node N+1 is an infinite bus, with E.sub.n+1 =0 and .delta..sub.n+i =0. Thus the voltage at node n+1 serves as a synchronous reference.
Equations 2a, 2b, 2c represent the post-fault condition. Suppose there is a set of (i elements) stable equilibrium points for the system described by 2a, 2b, 2c that is pairs of .delta..sub.i, w.sub.i, where w.sub.i is zero. Then, if it is assumed for purposes of this description that the transfer conductances are zero, a version of equations 2a, 2b, 2c can be derived where the stability region can be estimated: ##EQU2##
Using Lyapunov theory, a direct method of determining the transient stability of the post-fault condition can be deduced.
Let (.delta..sub.s, w.sub.s) be a stable post-fault equilibrium. Define the energy function V(.delta., w) of the state: ##EQU3##
At the post-fault equilibrium, V(.delta..sub.s, w.sub.s)=0. Differentiating V(.delta..sub.s,w.sub.s) along the trajectories of equation 3 gives ##EQU4## This inequality shows that V qualifies for use as a Lyapunov function since V(.delta.(t),w(t)) must decline along the post-fault trajectory.
Lyapunov theory then leads to the following approach to finding if the post-fault condition is converging toward stability: